Fourier finite-difference method and proper orthogonal decomposition for the quick spectral study of 2D periodic photonic structures

We present an original rigorous numerical method for calculating light diffraction by three-dimensional photonic structures with a two-dimensional periodicity. It is based on Fourier expansions in the two periodic directions and an elegant yet efficient combination of finite-difference discretization and modal treatment in the propagation direction. The Fourier finite-difference method (FFDM) is cast into an easy-to-handle, agile, and efficient matrix formalism. It uses matrix inversions instead of eigenmodes computations, a strategy that confers a clear advantage in terms of computation time for the large Fourier truncation orders that are typically used to deal with micro- and nanophotonic structures of current interest. The computational gain increases with the Fourier truncation order. It is maximal (∼30) for structures with subwavelength thicknesses, since a small number of finite-difference steps is sufficient to achieve a relative accuracy of approximately 1%. Our approach is particularly well suited for the analysis of generalized grating structures and metasurfaces. A second advantage of the proposed method is the possibility of using model-order reduction techniques. We use proper orthogonal decomposition (POD) to further speed up the computation of diffraction spectra as a function of the frequency, the incident angle, or a geometric parameter.